【正文】 an, variance and standard deviation are calculated. In this calculation, niyfifth percentile estimates of optimistic and pessimistic duration times are assumed, so that Equation () is applied. The critical path for this project ignoring uncertainty in activity durations consists of activities A, C, F and I as found in Table 103 (Section ). Applying the PERT analysis procedure suggests that the duration of the project would be approximately normally distributed. The sum of the means for the critical activities is + + + = days, and the sum of the variances is + + + = leading to a standard deviation of days. With a normally distributed project duration, the probability of meeting a project 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。 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。11. Advanced Scheduling Techniques Use of Advanced Scheduling Techniques Construction project scheduling is a topic that has received extensive research over a number of decades. The previous chapter described the fundamental scheduling techniques widely used and supported by numerous mercial scheduling systems. A variety of special techniques have also been developed to address specific circumstances or problems. With the availability of more powerful puters and software, the use of advanced scheduling techniques is being easier and of greater relevance to practice. In this chapter, we survey some of the techniques that can be employed in this regard. These techniques address some important practical problems, such as: ? scheduling in the face of uncertain estimates on activity durations, ? integrated planning of scheduling and resource allocation, ? scheduling in unstructured or poorly formul