【正文】 om variables, the variance or variation in the duration of this critical path is calculated as the sum of the variances along the critical path. With the mean and variance of the identified critical path known, the distribution of activity durations can also be puted. The mean and variance for each activity duration are typically puted from estimates of optimistic (ai,j), most likely (mi,j), and pessimistic (bi,j) activity durations using the formulas: () and () where and are the mean duration and its variance, respectively, of an activity (i,j). Three activity durations estimates (., optimistic, most likely, and pessimistic durations) are required in the calculation. The use of these optimistic, most likely, and pessimistic estimates stems from the fact that these are thought to be easier for managers to estimate subjectively. The formulas for calculating the mean and variance are derived by assuming that the activity durations follow a probabilistic beta distribution under a restrictive condition. [2] The probability density function of a beta distributions for a random varable x is given by: () 。 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。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 formulated circumstances. A final section in the chapter describes some possible improvements in the project scheduling process. In Chapter 14, we consider issues of puter based implementation of scheduling procedures, particularly in the context of integrating scheduling with other project management procedures. Back to top Scheduling with Uncertain Durations Section described the application of critical path scheduling for the situation in which activity durations are fixed and known. Unfortunately, activity durations are estimates of the actual time required, and there is liable to be a significant amount of uncertainty associated with the actual durations. During the preliminary planning stages for a project, the uncertainty in activity durations is particularly large since the scope and obstacles to the project are still undefined. Activities that are outside of the control of the owner are likely to be more uncertain. For example, the time required to gain regulatory approval for projects may vary tremendously. Other external events such as adverse weather, trench collapses, or labor strikes make duration estimates particularly uncertain. Two simple approaches to dealing with the uncertainty in activity durations warrant some discussion before introducing more formal scheduling procedures to deal with uncertainty. First, the uncertainty in activity durations may simply be ignored and scheduling done using the expected or most likely time duration for each activity. Since only one duration estimate needs to be made for each activity, this approach reduces the required work in setting up the original schedule. Formal methods of introducing uncertainty into the scheduling process require more work and assumptions. While this simple approach might be defended, it has two drawbacks. First, the use of expected activity durations typically results in overly optimistic schedules for pletion。 using a larger number of simulated durations would result in a more accurate estimate of the standard deviation. TABLE 113 Duration Realizations for a Monte Carlo Schedule Simulation Simulation Number Activity A Activity B Activity C