【正文】 er 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 schedule can be estimated from the results of a Monte Carlo simulation: ? Estimates of the expected time and variance of the project pletion. ? An estimate of the distribution of pletion times, so that the probability of meeting a particular pletion date can be estimated. ? The probability that a particular activity will lie on the critical path. This is of interest since the longest or critical path through the work may change as activity durations change. The disadvantage of Monte Carlo simulation results from the additional information about activity durations that is required and the putational effort involved in numerous scheduling applications for each set of simulated durations. For each activity, the distribution of possible durations as well as the parameters of this distribution must be specified. For example, durations might be assumed or estimated to be uniformly distributed between a lower and upper value. In addition, correlations between activity durations should be specified. For example, if two activities involve assembling forms in different locations and at different times for a project, then the time required for each activity is likely to be closely related. If the forms pose some problems, then assembling them on both occasions might take longer than expected. This is an example of a positive correlation in activity times. In application, such correlations are monly ignored, leading to errors in results. As a final problem and discouragement, easy to use software systems for Monte Carlo simulation of project schedules are not generally available. This is particularly the case when correlations between activity durations are desired. Another approach to the simulation of different activity durations is to develop specific scenarios of events and determine the effect on the overall project schedule. This is a type of whatif problem solving in which a manager simulates events that might occur and sees the result. For example, the effects of different weather patterns on activity durations could be estimated and the resulting schedules for the different weather patterns pared. One method of obtaining information about the range of possible schedules is to apply the scheduling procedure using all optimistic, all most likely, and then all pessimistic activity durations. The result is three project schedules representing a range of possible outes. This process of whatif analysis is similar to that undertaken during the process of construction planning or during analysis of project crashing. Example 111: Scheduling activities with uncertain time durations. Suppose that the nine activity example project shown in Table 102 and Figure 104 of Chapter 10 was thought to have very uncertain activity time durations. As a result, project scheduling considering this uncertainty is desired. All three methods (PERT, Monte Carlo simulation, and Whatif simulation) will be applied. Table 111 shows the estimated optimistic, most likely and pessimistic durations for the nine activities. From these estimates, the mean, 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