【正文】 these soil peculiarities might be responsible for the correlations of excavation activity durations described above. Results from the different methods are summarized in Table 112. Note that positive correlations among some activity durations results in relatively large increases in the expected project duration and variability. TABLE 112 Project Duration Results from Various Techniques and Assumptions for an Example Procedure and Assumptions Project Duration (days) Standard Deviation of Project Duration (days) Critical Path Method PERT Method Monte Carlo Simulation No Duration Correlations Positive Duration Correlations Whatif Simulations Optimistic Most Likely Pessimistic NA NA NA NA Back to top Calculations for Monte Carlo Schedule Simulation In this section, we outline the procedures required to perform Monte Carlo simulation for the purpose of schedule analysis. These procedures presume that the various steps involved in forming a work plan and estimating the characteristics of the probability distributions for the various activities have been pleted. Given a plan and the activity duration distributions, the heart of the Monte Carlo simulation procedure is the derivation of a realization or synthetic oute of the relevant activity durations. Once these realizations are generated, standard scheduling techniques can be applied. We shall present the formulas associated with the generation of normally distributed activity durations, and then ment on the requirements for other distributions in an example. To generate normally distributed realizations of activity durations, we can use a two step procedure. First, we generate uniformly distributed random variables, ui in the interval from zero to one. Numerous techniques can be used for this purpose. For example, a general formula for random number generation can be of the form: () where = and ui1 was the previously generated random number or a preselected beginning or seed number. For example, a seed of u0 = in Eq. () results in u1 = , and by applying this value of u1, the result is u2 = . This formula is a special case of the mixed congruential method of random number generation. While Equation () will result in a series of numbers that have the appearance and the necessary statistical properties of true random numbers, we should note that these are actually pseudo random numbers since the sequence of numbers will repeat given a long enough time. With a method of generating uniformly distributed random numbers, we can generate normally distributed random numbers using two uniformly distributed realizations with the equations: [3] () with where xk is the normal realization, x is the mean of x, x is the standard deviation of x, and u1 and u2 are the two uniformly distributed random variable realizations. For the case in which the mean of an activity is days and the standard deviation of the duration is days, a corresponding realization of the duration is s = , t = and xk = days, using the two uniform random numbers generated from a seed of above. Correlated random number realizations may be generated making use of conditional distributions. For example, suppose that the duration of an activity d is normally distributed and correlated with a second normally distributed random variable x which may be another activity duration or a separate factor such as a weather effect. Given a realization xk of x, the conditional distribution of d is still normal, but it is a function of the value xk. In particular, the conditional mean ( 39。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 anoth